An Improved Effective Gravitational Potential for Supernova Simulations

A&A 445, 273–289 (2006) Astronomy DOI: 10.1051/0004-6361:20052840 & c ESO 2005 Astrophysics

Exploring the relativistic regime with Newtonian hydrodynamics: an improved effective gravitational potential for supernova simulations

A. Marek, H. Dimmelmeier, H.-Th. Janka, E. Müller, and R. Buras

Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany e-mail: [email protected]

Received 8 February 2005 / Accepted 4 July 2005

ABSTRACT

We investigate the possibility approximating relativistic effects in hydrodynamical simulations of stellar core collapse and post-bounce evolution by using a modified gravitational potential in an otherwise standard Newtonian hydrodynamic code. Different modifications of a previously introduced effective relativistic potential are discussed. Corresponding hydrostatic solutions are compared with solutions of the TOV equations, and hydrodynamic simulations with two different codes are compared with fully relativistic results. One code is applied for one- and two-dimensional calculations with a simple equation of state, and employs either the modified effective relativistic potential in a Newtonian framework or solves the general relativistic field equations under the assumption of the conformal flatness condition (CFC) for the three-metric. The second code allows for full-scale supernova runs including a microphysical equation of state and neutrino transport based on the solution of the Boltzmann equation and its moments equations. We present prescriptions for the effective relativistic potential for self-gravitating fluids to be used in Newtonian codes, which produce excellent agreement with fully relativistic solutions in spherical symmetry, leading to significant improvements compared to previously published approximations. Moreover, they also approximate qualitatively well relativistic solutions for models with rotation.

Key words. gravitation – hydrodynamics – methods: numerical – relativity – stars: supernovae: general

1. Introduction utilizes a generalised potential to approximate relativistic gravity. The Agile-BoltzTran code of the Oak Ridge-Basel It is a well known fact that gravity plays an important role group (Liebendörfer et al. 2001, 2002, 2004, 2005) is a fully during all stages of a core collapse supernova. Gravity is the relativistic (1D) hydrodynamics code. The comparison showed driving force that at the end of the life of massive stars over- that both codes produce qualitatively very similar results except comes the pressure forces and causes the collapse of the stellar for some small (but growing) quantitative differences occurring core. Furthermore, the subsequent supernova explosion results in the late post-bounce evolution. Inspired by this comparison from the fact that various processes tap the enormous amount we explored improvements of the effective relativistic potential of gravitational binding energy released during the formation of used by Rampp & Janka (2002) in order to achieve an even bet- ff the proto-neutron star. General relativistic e ects are important ter agreement than that reported by Liebendörfer et al. (2005). for this process and cannot be neglected in quantitative mod- To this end we tested different variants of approximations to els because of the increasing compactness of the proto-neutron relativistic gravity employing two simulation codes. star. Therefore, it is important to include a proper treatment of general relativity or an appropriate approximation in the nu- On the one hand purely hydrodynamic simulations were merical codes that one uses to study core collapse supernovae. performed with the code CoCoNuT of Dimmelmeier et al. Recently Liebendörfer et al. (2005) performed a compar- (2002a, 2005) assuming spherical or axial symmetry. This code ison of the results obtained with the supernova simulation optionally either uses a Newtonian (or alternatively an effec- codes Vertex and Agile-BoltzTran which both solve the tive relativistic) gravitational potential in a Newtonian treat- Boltzmann transport equation for neutrinos. The Vertex code ment of the hydrodynamic equations or solves the general rel- (see Rampp & Janka 2002) is based on the Newtonian hy- ativistic equations of fluid dynamics and the relativistic field drodynamics code Prometheus (Fryxell et al. 1990) and equations. The latter are formulated under the assumption of the conformal flatness condition (CFC) for the three-metric, Appendices A and B are only available in electronic form at also known as the Isenberg–Wilson–Mathews approximation http://www.edpsciences.org (Isenberg 1978; Wilson et al. 1996), which is identical to

Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.doi.org/10.1051/0004-6361:20052840 274 A.Mareketal.:Animprovedeffective potential for supernova simulations solving the exact general relativistic equations in spherical by the TOV potential symmetry. This allows for a direct comparison of different ef- ∞ dr mTOV 3 fective relativistic potentials with a fully relativistic treatment Φ (r) = −4π + r (P + pν) TOV 2 π in both spherically symmetric and axisymmetric simulations r r 4 1 ρ + e + P using the same code. × , (2) On the other hand, we used the computationally expensive Γ2 ρ Vertex code (Rampp & Janka 2002) for spherically symmet- to obtain the effective relativistic potential Φ ff as ric supernova simulations including neutrino transport and a e microphysical equation of state (EoS). The results of these cal- Φeff =ΦTOV. (3) culations performed with different effective relativistic poten- ρ = ρ tials are compared with those obtained with the general rela- Here is the rest-mass density, e is the internal energy tivistic Agile-BoltzTran code of the Oak Ridge-Basel group density with being the specific internal energy, and P is the (cf. Liebendörfer et al. 2005). gas pressure. The TOV mass is given by In this paper we discuss the results of our test calculations r v = π 2 ρ + + + F , and present improved effective relativistic potentials which are mTOV(r) 4 dr r e E (4) 0 Γ simple to implement without modifying the Newtonian equations of hydrodynamics in an existing code, and which approx- where pν, E,andF are the neutrino pressure, the neutrino en- imate the results of general relativistic simulations very well. ergy density, and the neutrino flux, respectively (Baron et al. The paper is organised as follows: in Sect. 2 we describe 1989; Rampp & Janka 2002). v the effective relativistic potentials used in this investigation. In The fluid velocity is identified with the local radial veloc- Γ Sect. 3 we briefly discuss the most important features of the ity calculated by the Newtonian code and the metric function numerical codes we used for our hydrodynamic simulations. In is given by Sect. 4 we present the results obtained by applying the effec- 2m tive relativistic potentials in spherically symmetric simulations Γ= 1 + v2 − TOV · (5) of supernova core collapse and to neutron star models, while r Sect. 5 is devoted to a discussion of the multi-dimensional The velocity-dependent terms were added for a closer match simulations of rotational supernova core collapse. Finally, in with the general relativistic form of the equation of motion Sect. 6 we summarise our findings and draw some conclusions. (van Riper 1979; Baron et al. 1989). In the treatment of neu- Throughout the article, we use geometrised units with c = trino transport general relativistic redshift and time dilation G = 1. effects are included, but for reasons of consistency with the Newtonian hydrodynamics part of the code the distinction between coordinate and proper radius is ignored in the relativistic 2. Effective relativistic potential transport equations (for details, see Sect. 3.7.2 of Rampp & Janka 2002). The quality of this approach was ascertained by a ff Approximating the e ects of general relativistic gravity in a comparison with fully relativistic calculations (Rampp & Janka Newtonian hydrodynamics code may be attempted by using an 2002; Liebendörfer et al. 2005). ff Φ e ective relativistic gravitational potential eff which mimics In order to calculate the effective relativistic potential for the deeper gravitational well of the relativistic case. In the fol- multi-dimensional flows we substitute the “spherical contribu- ff lowing Sects. 2.1 and 2.2, several of these e ective relativistic tion” Φ (r) to the multi-dimensional Newtonian gravitational potentials will be discussed. potential ρ Φ ,θ,ϕ = − θ ϕ 2 θ 2.1. TOV potential for a self-gravitating fluid (r ) dr d d r sin (6) V |r − r | For a self-gravitating fluid it is desirable that an effective rel- by the TOV potential Φ : ativistic potential reproduces the solution of hydrostatic equi- TOV ff librium according to the Tolman–Oppenheimer–Volko (TOV) Φeff =Φ− Φ+Φ TOV. (7) equation. With this requirement in mind and comparing the rel- Φ Φ ativistic equation of motion (cf. van Riper 1979; Baron et al. Here (r)and TOV are calculated according to Eqs. (1) 1989) with its Newtonian analogon, Rampp & Janka (2002) re- and (2), respectively, however with the hydrodynamic quanti- arranged the relativistic terms into an effective relativistic po- ties ρ, e, P, v and the neutrino quantities E, F, pν being replaced tential (see Kippenhahn & Weigert 1990, for the hydrostatic, by their corresponding angularly averaged values. Note that v neutrino-less case). here refers to the radial component of the velocity, only. Thus for spherically symmetric simulations using a Newtonian hydrodynamics code the idea is to replace the 2.2. Modifications of the TOV potential Newtonian gravitational potential In a recent comparison Liebendörfer et al. (2005) found that ∞ ρ gravity as described by the TOV potential in Eq. (3) overrates Φ = − π 2 (r) 4 dr r | − | (1) the relativistic effects, because in combination with Newtonian 0 r r A.Mareketal.:Animprovedeffective potential for supernova simulations 275 kinematics it tends to overestimate the infall velocities and to per nucleon – both appear in Eqs. (2), (4) only combined in underestimate the flow inertia in the pre-shock region. Thus, form of the “relativistic energy” per unit of mass, (ρ + e)/ρ –is supposedly via the nonlinear dependence of Φeff on e and P well defined, but not the individual parts. Therefore there exists the compactness of the proto-neutron star is overestimated, ambiguity with respect to which contribution to the energy is with this tendency increasing at later times after core bounce. set to zero. In order to assess a possible sensitivity of the core Consequently, the neutrino luminosities and the mean energies collapse results to this ambiguity, we tried two different vari- of the emitted neutrinos are larger than in the corresponding ants of Case B in our Vertex simulations with microphysical relativistic simulation. EoS. On the one hand we used e = E = F = 0 in Eq. (4), In order to reduce these discrepancies – without sacrific- with e being the internal energy density plus an energy nor- ing the simplicity of Newtonian dynamics – we tested several malization given by the EoS of Lattimer & Swesty (1991), e = −24 modifications of the TOV potential, Eqs. (2), which all act to (ρ+e)−ρ (mn −∆)/mu (where mu = 1.66×10 g is the atomic weaken it. In particular, we studied the following variations: mass unit, mn the neutron rest mass, and ∆=8.8MeV).Onthe other hand we tested e∗ = E = F = 0 with e∗ being e without Γ ∗ Case A: In the integrand of Eq. (4) a factor , Eq. (5) is added. this energy normalization, i.e. e = (ρ+e)−ρ mn/mu. Nucleons Since Γ < 1 this reduces the gravitational TOV mass used are then assumed to contribute to the TOV mass, Eq. (4), with in the potential. the vacuum rest mass of the neutron, increasing the mass integral and reducing Γ, Eq. (5), relative to the first case, thus Case B: In Eq. (4) the internal gas energy density and the making the effective relativistic potential a bit stronger again. neutrino terms are set to zero, e = E = F = 0, which again In order to compensate for this we also set pν = 0intheTOV decreases the gravitational TOV mass. potential, Eq. (2). Both variants are found to yield extremely Case C: In Eq. (2) the internal gas energy is set to zero, e = 0, similar results and we therefore will discuss only one of them which directly weakens the TOV potential. (the first variant) as Case B for the Vertex simulations. ff Case D: In the equation for the TOV potential, Eq. (2), Ideally, a Newtonian simulation with an e ective relativis- 1 + tic potential not only yields a solution of the TOV structure mTOV is replaced by 2 (mTOV mg). Here a Newtonian = − equations for an equilibrium state (as does Case R), but in ad- gravitational mass is defined as mg mr mb with the rest r m = π r r2ρ dition closely reproduces the results from a relativistic simu- mass r 4 0 d and the mass equivalent of the r lation (Case GR) during a dynamic evolution. Applying the binding energy m = 2π| dr r2ρ Φ|.Asm < m ,the b 0 g TOV modifications of the TOV potential listed above, we find that strength of the potential is reduced. Cases A to D yield improved results as compared to Case R, Case E: Both in the equation for the TOV potential, Eq. (2), while Cases E to G either weaken the potential too much or are and the equation for the TOV mass, Eq. (4), we set e = 0. very close to Case R. These findings are detailed in Sects. 4 and 5. Case F: In the equation for the TOV potential, Eq. (2), we set Γ=1. As Γ < 1 otherwise, this weakens the potential. Case G: In the expression for Γ, Eq. (5), the velocity is set to 2.3. Theoretical motivation v = Γ−2 zero, 0. Hence, increases in Eq. (2). This modifi- There are (at least) two basic requirements which appear de- cation is used to also test a potential which is even stronger sirable for an effective relativistic potential in a Newtonian than the unmodified TOV potential. simulation. Firstly, the far field limit of the fully relativistic treatment should be approximated reasonably well in order to In addition to these cases with a modified version of the TOV follow the long-term accretion of the neutron star and the as- potential, we use the following notations: sociated growth of its baryonic mass. Secondly, the hydrostatic structure of the neutron star should well fit the solution of the Case N: This denotes the purely Newtonian runs with “regu- TOV equations. lar” Newtonian potential. The second point will be discussed in detail in Sect. 4.1. Case R: This is the “reference” case with the TOV potential A closer consideration of the first point suggests the modified as defined by Eq. (2). effective relativistic potential of Case A as promising, and in Case GR: This case refers to fully relativistic simulations with fact it turns out to be the most preferable choice concerning either the code CoCoNuTortheAgile-BoltzTran neu- consistency and quality of the results. The other cases listed trino radiation-hydrodynamics code of the Oak Ridge- in Sect. 2.2 are mostly ad hoc modifications of the original ef- Basel collaboration. fective relativistic potential of Eqs. (2)–(4) (Case R) with the aim to reduce its strength, which was found to overestimate the Note that setting the internal energy density e to zero in Case B effects of gravity compared to fully relativistic simulations in is unambiguous when a simple EoS is used and the particle previous work (Liebendörfer et al. 2005). These cases are also rest masses are conserved. In general, however, particles can be discussed here for reasons of comparison and completeness. created and destroyed, or bound states can be formed (e.g., in In Eq. (4) the hydrodynamic quantities (like rest-mass pair annihilation processes or nuclear reactions, respectively). density ρ plus extra terms) are integrated over volume. In Then only the sum of the rest mass energy and internal energy the Newtonian treatment there is no distinction between 276 A.Mareketal.:Animprovedeffective potential for supernova simulations coordinate volume and local proper volume. Performing the in- The arguments given above only provide a heuristic justi- tegral of Eq. (4) therefore leads to a mass – used as the mass fication for the manipulation of the TOV potential proposed in which produces the gravitational potential in Eqs. (2) and (3) – Case A. A deeper theoretical understanding and more rigorous 2 which is larger than the baryonic mass, mb = 4π dr r ρ.In analytical analysis of its consequences and implications is cer- particular, it is also larger than the gravitational mass in a con- tainly desirable, but beyond the scope of the present paper. We sistent relativistic treatment, which is the volume integral of plan to return to this question in future work. the total energy density and includes the negative gravitational potential energy of the compact object. The latter reduces the 3. Numerical methods gravitational mass relative to the baryonic mass by the gravitational binding energy of the star (see, e.g., Shapiro & Teukolsky In the following we briefly introduce the two numerical codes 1983, p. 125 for a corresponding discussion). Therefore, the used for the simulations presented in this work. The codes are effective relativistic potential introduced by Rampp & Janka based on state-of-the-art numerical methods for hydrodynam- (2002) (our Case R, Eqs. (2)–(4)) cannot properly reproduce ics and neutrino transport coupled to gravity, and were used to the far field limit of the relativistic case and thus overestimates simulate stellar core collapse and supernova explosions. the effects of gravity. This particularly applies to the infall velocities of the stellar gas ahead of the supernova shock, as shown in Liebendörfer et al. (2005). 3.1. Hydrodynamic code without transport Introducing an extra factor Γ in the integral of Eq. (4) for The purely hydrodynamic simulations are performed with the the TOV mass is motivated by the following considerations code CoCoNuT developed by Dimmelmeier et al. (2002a,b) (where for reasons of simplicity contributions from neutrinos, with a metric solver based on spectral methods as described though important, are neglected and spherical symmetry is as- in Dimmelmeier et al. (2005). The code optionally uses ei- sumed): In the relativistic treatment the total (gravitating) mass ther a Newtonian gravitational potential (or an effective rela- of the star is given as tivistic potential, see Sect. 2), or the general relativistic field ∞ equations for a curved spacetime in the ADM 3 + 1-split un- 2 mg = 4π dr r (ρ + e), (8) der the assumption of the conformal flatness condition (CFC) 0 for the three-metric. The (Newtonian or general relativistic) hy- where dV = 4πdr r2 is the coordinate volume element, drodynamic equations are consistently formulated in conserva- whereas the baryonic mass is tion form, and are solved by high-resolution shock-capturing schemes based upon state-of-the-art Riemann solvers and third- ∞ 2 −1 order cell-reconstruction procedures. Neutrino transport is not mb = 4π dr r Γ ρ, (9) 0 included in the code. A simple hybrid ideal gas EoS is used that consists of a polytropic contribution describing the degen- − with dV = 4πdr r 2Γ 1 being the local proper volume ele- erate electron pressure and (at supranuclear densities) the pres- Γ−1 > ment. The factor 1intheintegrandofmb thus ensures sure due to repulsive nuclear forces, and a thermal contribution > that mb mg. The integral for mg can also be written as which accounts for the heating of the matter by shocks: ∞ Γ ∞ 2 P = Pp + Pth, (12) mg = 4π dr r (ρ + e) = dV Γ (ρ + e). (10) 0 Γ 0 where Since in Newtonian hydrodynamics no distinction is made be- γ tween coordinate and proper volumes, one may identify dV ≡ Pp = Kρ , Pth = ρth(γth − 1), (13) dV, consistent with the rest of our Newtonian code. This leaves and = − . The polytropic specific internal energy is the additional factor Γ in the integrand of Eq. (10), leading to a th p p determined from P by the ideal gas relation in combination redefined TOV mass used for computing the effective relativis- p with continuity conditions in the case of a discontinuous γ.In tic potential in Case A, that case, the polytropic constant K also has to be adjusted (for r v more details, see Dimmelmeier et al. 2002a; Janka et al. 1993). = π 2Γ ρ + + + F · mTOV(r) 4 dr r e E (11) The CoCoNuT code utilizes Eulerian spherical coordi- 0 Γ nates {r,θ,ϕ}, and thus axially or spherically symmetric con- − The fact that a factor Γ 1 in the volume integral establishes the figurations can be easily simulated. For the core collapse sim- relation between gravitating mass, Eq. (8), and baryonic mass, ulations discussed in Sects. 4.1 and 5, the finite difference grid Eq. (9), in the relativistic case suggests that the factor Γ < 1in consists of 200 logarithmically spaced radial grid points with a Eq. (11) might lead to a suitable reduction of the overestimated central resolution of 500 m. A small part of the grid covers an effective potential that results when the original TOV mass of artificial low-density atmosphere extending beyond the core’s Eq. (4) is used in Eqs. (2) and (5). Indeed, a comparison of the outer boundary. The spectral grid of the metric solver is split integral of Eq. (11) for large r with the rest-mass energy of a into 6 radial domains with 33 collocation points each. In or- neutron star reduced by its binding energy at time t (computed der to be able to accurately follow the dynamics, the domain t t Lν t Lν from the emitted neutrino energy, 0 d ( ) with being the boundaries adaptively contract towards the centre during the neutrino luminosity) reveals very good agreement. collapse, as described in Dimmelmeier et al. (2005). For the A.Mareketal.:Animprovedeffective potential for supernova simulations 277 migration test (Sect. 4.3) the finite difference grid consists of for the numerical setup (e.g., the grids for hydrodynamics and 250 logarithmically spaced radial grid points, of which 60 and neutrino transport). Information about this setup can be found 190 cover the neutron star and the atmosphere, respectively. in Liebendörfer et al. (2005). The initial model for our calcula- For the two-dimensional simulations of rotational core collapse tions is the 15 M progenitor model “s15s7b2” from Woosley (Sect. 5) the finite difference and spectral grids are extended by & Weaver (1995). 30 and 17 equidistant angular grid points, respectively. Since solving the neutrino transport problem is computa- Even when using spectral methods the calculation of the tionally quite expensive we performed calculations only for spacetime metric is computationally expensive. Hence, in Cases A, B, and F (as defined in Sect. 2.2) with the Vertex the relativistic spherical (rotational) core collapse simulations code. The quality of these results is then compared to the fully the metric is updated only once every 10/1/50 (100/10/50) hy- relativistic treatment of the Agile-BoltzTran code. drodynamic time steps before/during/after core bounce, and extrapolated in between. In the migration test the update is performed every 10th step during the entire evolution. The nu- 3.3. Hydrodynamics and implementation merical adequacy of this procedure is tested and discussed in of the effective relativistic potential detail in Dimmelmeier et al. (2002a). The implementation of an effective relativistic gravitational The quality of the CFC approximation has been tested com- potential or of an effective relativistic gravitational force as prehensively in the context of supernova core collapse and for its derivative into existing Newtonian hydrodynamics codes neutron star models (Cook et al. 1996; Dimmelmeier et al. is straightforward and does not differ from the use of the 2002a; Shibata & Sekiguchi 2004; Dimmelmeier et al. 2005; Newtonian potential or force. Cerdá-Durán et al. 2005). For single stellar configurations the The equations solved by the two codes used for our sim- CFC approximation is very accurate as long as neither the com- ulations are described in much detail in previous publications pactness of the star is extremely relativistic nor its rotation is (see, e.g., Rampp & Janka 2002; Müller & Steinmetz 1995; ff too fast and di erential. Both criteria are well fulfilled during Dimmelmeier et al. 2002a, 2005, and references therein). The supernova core collapse. Concerning core collapse simulations implementation of the source terms of the gravitational poten- ff of rapidly and strongly di erentially rotating configurations at tial as discussed in Müller & Steinmetz (1995) is also applied very high densities, we recently found that the CFC approx- for the handling of the effective relativistic potentials investi- imation yields excellent agreement with formulations solving gated in this work. The only specific feature here is the mutual the exact spacetime metric even in this extreme regime (Ott dependence of Γ and mTOV, which is either accounted for by a et al. 2005). rapidly converging iteration or by taking Γ from the old time As the three-metric of any spherically symmetric spacetime step in the update of mTOV, when the changes during the time can be written in a conformally flat way, the spherical CFC steps are sufficiently small. simulations presented in Sects. 4.1 and 4.3 involve no approx- Since the actual form of the gravitational source term is imation, i.e., the exact Einstein equations are solved in these unspecified in the conservation laws of fluid dynamics, the cases. Newtonian potential can be replaced by the effective relativistic potentials investigated in the current work in a technically 3.2. Hydrodynamic code with neutrino transport straightforward way. Solving an equation for the internal plus kinetic energy (as in our codes) requires a treatment of the grav- The Vertex neutrino-hydrodynamics code solves the conser- ity source term in this equation that is consistent with its imple- vation equations of Newtonian hydrodynamicsin their Eulerian mentation in the equation of momentum. and conservative form using the PPM-based Prometheus Of course, the effective potential must be investigated code (Fryxell et al. 1990). The neutrino transport is treated concerning its consequences for the conservation of momen- by determining iteratively a solution of the coupled set of mo- tum and energy. Since a potential constructed according to ments equations and Boltzmann equation, achieving closure by Eqs. (2), (4) does not satisfy the Poisson equation, the momen- a variable Eddington factor method (Rampp & Janka 2002). tum equation cannot be cast into a conservation form (cf. Shu See the latter reference also for details about the coupling be- 1992, Part I, Chapter 4). As a consequence, the total linear mo- tween the hydrodynamics and neutrino transport parts of the mentum is strictly conserved only when certain assumptions Vertex code. The models presented in this paper are com- about the symmetry of the matter distribution are made, for ex- puted with the equation of state of Lattimer & Swesty (1991), ample in the case of spherical symmetry or axially symmetric in agreement with the choice of input physics in the work configurations with equatorial symmetry, or when only one oc- of Liebendörfer et al. (2005), where results obtained with the tant is modeled in the three-dimensional case. In axisymmetric Vertex code were compared with those from the Newtonian simulations the conservation of specific angular momentum is and fully relativistic calculations with the Agile-BoltzTran fulfilled as well, when using the effective relativistic potential. code of the Oak Ridge-Basel collaboration. In general, however, a sufficient quality of momentum (and an- In order to compare the results presented here with those gular momentum) conservation has to be verified by inspecting of the calculations of Liebendörfer et al. (2005) we used the the numerical results. same set of neutrino interaction rates as picked for Model G15 The long-range nature of gravity prohibits to have an equa- in Liebendörfer et al. (2005), and exactly the same parameters tion in pure conservation form for the total energy, i.e., for the 278 A.Mareketal.:Animprovedeffective potential for supernova simulations sum of internal, kinetic, and gravitational energy (Shu 1992, 4

Part I, Chapter 4). In contrast to the Newtonian case, however, ] ﬀ -3 90.5 ms our e ective relativistic potential does also not allow one to de- 3 90.1 ms 140.0 ms

rive a conservation equation for the total energy integrated over g cm all space. Monitoring global energy conservation in a simula- 14 2 ρ nuc

ff [10 tion with e ective relativistic potential therefore requires inte- c ρ 1 gration of the gravitational source terms over all cells in time. If 80.0 ms ff the local e ects of relativistic gravity are convincingly approx- 0 imated – as measured by good agreement with static solutions 80 90 100 110 120 130 140 of the TOV equation and with fully relativistic, dynamical sim- t [ms] ulations – there is confidence that the integrated action of the employed gravitational source term approximates well also the 0.0 80.0 ms global conversion between total kinetic, internal, and gravitational energies found in a relativistic simulation. -0.1 ] 90.1 ms c

The recipes for approximating general relativity should be [

v 140.0 ms applicable equally well in hydrodynamic codes different from -0.2 our (Eulerian) PPM schemes, provided the effects of gravity are -0.3 consistently treated in the momentum and energy equations. 90.5 ms ff The proposed e ective potentials are intended to yield a good 1 10 100 1000 ff representation of the e ects of relativistic gravity in particular r [km] in the context of stellar core collapse and neutron star forma- ρ tion. For our approximation to work well, the fluid flow should Fig. 1. Time evolution of the central density c (top panel) and radial be subrelativistic. The numerical tests described in the follow- profiles of the velocity v (bottom panel) for the accretion shock model ing sections show that velocities up to about 20% of the speed AS in relativistic gravity (Case GR). The velocity profiles are snap- of light are unproblematic. shots long before bounce, during bounce shortly before and after maximum density, and long after bounce, respectively. At t = 140.0msthe prompt shock has turned into an accretion shock. 4. Simulations in spherical symmetry 4.1. Supernova core collapse with simplified equation an accretion shock shortly after core bounce, we choose the of state EoS parameters γ1 = 1.325, γ2 = 3.0, and γth = 1.2inourac- We first test the quality of the TOV potential and its various cretion shock model AS (Fig. 1) to reproduce such a behaviour. modifications as an effective relativistic potential in the moder- The conversion of the prompt shock into an accretion shock at ately relativistic regime by evolving models of supernova core late times is reflected by the radial velocity profiles plotted in collapse with the simple EoS described in Sect. 3.1. Using the the bottom panel of Fig. 1. CoCoNuT code we are able to perform a direct comparison When recalculating Model AS with regular Newtonian with a Newtonian version of the code (employing either the gravity (Case N), the central density remains below the one for Newtonian, TOV, or modified TOV potential) against the rela- relativistic gravity (Case GR) during the entire evolution (see tivistic version in a curved spacetime. top panel of Fig. 2). This behaviour was already observed in The initial models are polytropes, which mimic an iron the rotational core collapse simulations of Dimmelmeier et al. core supported by electron degeneracy pressure, with a cen- (2002a). Replacing the regular Newtonian potential (Case N) 10 −3 ff tral density ρci = 10 gcm and EoS parameters γi = 4/3 by the TOV potential (Case R) yields the opposite e ect, and K = 4.897 × 1014 (in cgs units). To initiate the collapse, i.e., the central density is too high throughout the evolution. the initial adiabatic index is reduced to γ1 <γi.Atdensi- However, when using modifications A or B of the TOV poten- 14 −3 ρ ties ρ>ρnuc ≡ 2.0 × 10 gcm the adiabatic index is in- tial, the behaviour of c agrees very well with that observed > creased to γ2 ∼ 2.5 to approximate the stiffening of the EoS. in the relativistic simulation (Case GR), indicating that the dy- This leads to a rebound of the core (“core bounce”). The inner namics of the core collapse with this effective relativistic po- part of the core comes to a halt, while a prompt hydrodynamic tential is similar to that of the relativistic case. shock starts to propagate outward. Since the matter model of When applying the same methods to a model with a prompt the CoCoNuT code does not account for neutrino cooling, shock which continues to propagate, we observe the same the compact collapse remnant (corresponding to the hot proto- qualitative behaviour (see bottom panel of Fig. 2). For this neutron star) cannot cool and shrink, but for some models only Model PS the EoS parameters are set to γ1 = 1.300, γ2 = 2.0, slightly grows in mass due to accretion. These stages of the and γth = 1.5, respectively. Both during core bounce and at evolution can be seen in the top panel of Fig. 1, where the time late times (see magnifying inset) the Newtonian simulations evolution of the central density ρc is plotted for a typical core with modifications A and B of the TOV potential also yield collapse model. results which agree noticeably better with the relativistic one As it is known from numerical models with sophisticated (Case GR) than with the TOV potential (Case R), and obviously microphysics that the prompt hydrodynamic shock turns into also much better than with the regular potential (Case N). A.Mareketal.:Animprovedeffective potential for supernova simulations 279

5 4 Relativistic (Case GR) AS ] -3

] 3 -3 4 g cm

14 2 ρ

g cm nuc 14 [10 ρ 1 [10

c 3

ρ 0 4 Newtonian with regular potential (Case N) ]

2 -3 9 3 PS g cm

8 14 2 ] ρ nuc -3

7 [10 ρ 1 g cm

14 6 0 [10 c 5 4 Newtonian with TOV potential (Case R) ρ ] -3 4 3

3 g cm 0 5 10 14 2 ρ − nuc t tb [ms] [10 ρ 1

Fig. 2. Time evolution of the central density ρc for the accretion shock 0 model AS (top panel) and prompt shock model PS (bottom panel). 0 5 10 15 The regular Newtonian simulation (Case N; dashed line) yields a r [km] smaller ρc compared to the relativistic one (Case GR; solid line). If a TOV potential is used (Case R; dashed-dotted line), ρc is too high, Fig. 3. Comparison of radial profiles of the density ρ from the hydro- while modifications A (dotted line) and B (long-dashed line) of the dynamic evolution of Model AS at late times (solid line) and from a TOV potential give results close to the relativistic one. solution of the TOV structure equations with identical central density (dashed line). In both the relativistic simulation (Case GR; top panel) and the Newtonian simulation with TOV potential (Case R; bottom The time plotted in Fig. 2 is t − tb,wheretb is the time of panel) the two curves are almost indistinguishable, while the regular bounce for each simulation. We point out that the time coor- Newtonian simulation (Case N; centre panel) yields a large mismatch. dinate t in the relativistic spacetime is the time of an observer For comparison, the relativistic profile (solid line in the top panel) is being at rest at infinity, and thus we can directly relate it to also plotted in dashed-dotted line style in the lower two panels. the time used in the Newtonian simulation (with regular or effective relativistic potential). Moreover, at core bounce the differences between the relativistic coordinate time t and the = ρ, relativistic proper time tpc = αc dt of the fluid element at EoS, P P( ), defined by Eqs. (12), (13). Details of this the center, where α is the ADM lapse function, are negligi- procedure are given in Appendix A. The top panel of Fig. 3 ble for the gravitational fields encountered in our core collapse shows that this behaviour is indeed confirmed by our simula- models. In addition, shortly after core bounce the compact col- tions. The density profile obtained from a dynamic simulation lapse remnant reaches an approximate hydrostatic equilibrium with relativistic gravity (Case GR) and the one obtained from with nearly constant ρc. Therefore, for the models considered solving the TOV structure equations agree extremely well both here the time evolution of the central density in proper time tpc at supranuclear and subnuclear densities. Only ahead of the would look similar to the one in coordinate time t. outward propagating shock (off scale in Fig. 3) where matter The discussion up to now has only been concerned with the is still falling inward, i.e., where the radial velocity is not neg- local quantity ρc. While this is a good measure for the global ligible, a disagreement can be observed. collapse dynamics, a direct comparison of, e.g., radial profiles When using regular Newtonian gravity (Case N; see centre at certain evolution times is more powerful. panel of Fig. 3) the situation is different in two aspects. First, In Fig. 3 radial profiles of the density ρ for the core collapse due to the shallower Newtonian gravitational potential, the den- model AS are compared at “late” times (t = 150 ms, corre- sity is significantly smaller in the central parts of the compact sponding to about 60 ms after core bounce). As this model ex- remnant compared to relativistic gravity (Case GR). However, hibits no post-bounce mass accretion, shortly after core bounce beyond r ∼ 10 km the density in the Newtonian model exceeds the compact remnant has acquired a new equilibrium state that of the relativistic simulation, which is the density crossing with essentially vanishing radial velocity. Its density profile can effect observed and discussed in detail by Dimmelmeier et al. thus be very well described by that of a TOV solution con- (2002b). The weaker regular Newtonian potential thus yields structed with equal central density ρc and an “effective” one- a remnant which is less compact and dense in the centre, but parameter EoS, P = P(ρ), extracted from the two-parameter relatively denser in the outer parts. 280 A.Mareketal.:Animprovedeffective potential for supernova simulations

0.14 the TOV equation with equal central value and the proﬁle com- Relativistic (Case GR) puted with relativistic gravity (Case GR), and does not even 0.12 show the distinctive steps of that proﬁle. ] 2

c The above ﬁndings imply that the ﬁnal equilibrium state of ∋

[ a Newtonian core collapse simulation performed with the TOV 0.10 potential (Case R) is consistent with a TOV solution of identical central density and specific internal energy. However, if rel- 0.08 ativistic effects are important, the density stratification and en- Newtonian with regular potential (Case N) ergy distribution (which in turn determines the “effective” EoS as described in Appendix A) will differ from that of a consis- 0.10 ]

2 tently relativistic simulation (Case GR). Presumably the com-

c ∋

[ bination of the relativistic TOV potential (which depends non- linearly on the pressure and the internal energy density) with 0.05 Newtonian kinematics in Case R is responsible for this differ- 0.14 ence in the stratification of density and specific internal energy Newtonian with TOV potential (Case R) in the compact remnant compared to Case GR (as seen in the bottom panels of Figs. 3 and 4). Currently we cannot provide 0.12

] a more detailed analysis for understanding the mechanism by 2

c ∋ which these discrepancies are caused. We plan to conduct fur- [ 0.10 ther research on this issue. As discussed in Sect. 2.2, a modification of a Newtonian simulation incorporating the effects of relativistic gravity must 0.08 0 5 10 15 not only yield equilibrium states which (approximately) repro- r [km] duce a solution of the TOV equations, particularly for compact matter configurations. More importantly, the density and en- Fig. 4. Same as Fig. 3, but for radial profiles of the specific internal energy . The step-like shape of the profile arises from contributions ergy profiles must also agree as closely as possible with the profiles obtained from the evolution with a relativistic code of the thermal energy th in various strengths. (which implies the previous requirement). Both these requirements are fulfilled for Newtonian simulations utilizing modifi- Second, when comparing the density profile resulting from cation A and B of the TOV potential, as shown for profiles of the hydrodynamic evolution with that obtained by solving the the density and specific internal energy in the final equilibrium TOV structure equations (for the same ρc), a clear disagree- state of the core collapse model AS in Figs. 5 and 6, respec- ment can be noticed. As the core reaches a final compact- tively. In these cases the profiles agree well with the relativistic ness 2Mcore/Rcore ∼ 0.2 as measured by the ratio of core result (Case GR) not only near the centre (as already observed mass to core radius, a hydrodynamic simulation with regular in the time evolution of ρc; see top panel of Fig. 2) but also Newtonian potential cannot be expected to yield an equilibrium throughout the entire core, with the agreement being much bet- result which is identical to a solution of the (relativistic) TOV ter compared to the TOV potential (Case R). Accordingly, the structure equations. solution of the TOV structure equations for identical central If on the other hand the Newtonian hydrodynamic simula- values yields similar profiles as well. tion is performed with the TOV potential (Case R; see bottom We conclude that the modifications A and B of the TOV panel of Fig. 3), the final equilibrium state is (as in Case GR) potential combined with Newtonian kinematics provide a very a solution of the TOV structure equations. However, after core good approximation of the results obtained with relativistic bounce the density profile of Case R lies well above that ob- gravity (Case GR) in the strongly dynamic phases during core tained in the relativistic simulation (Case GR). Thus Case GR bounce as well as at post-bounce times when the compact rem- and Case R both exhibit post-bounce density stratifications nant acquires equilibrium. The same findings hold for the mod- which are consistent with a relativistic equilibrium state but ifications C and D of the TOV potential, although we do not otherwise differ noticeably from each other. present results for these cases here. On the other hand, com- Radial profiles of the specific internal energy (see Fig. 4) pared to Cases A to D, the modifications E to G produce less exhibit a similar behaviour as the density profiles presented in accurate results. Thus, we consider them inappropriate for an Fig. 3. As expected, both the relativistic simulation (Case GR) effective relativistic potential. and the Newtonian simulation with TOV potential (Case R) In passing we note that gauge dependences may play an yield final equilibrium profiles which closely coincide with the important role when non-invariant quantities (like, e.g., density corresponding solution of the TOV structure equations (see top profiles) from simulations with a relativistic code are compared and bottom panel, respectively). Again, analogous to the den- against results from either a relativistic code using a different sity, in Case R the internal energy in the compact remnant is formulation of the spacetime metric or from a Newtonian code. significantly higher as compared to Case GR. In the Newtonian For instance, the radial coordinate r used in the profile plots in simulation with regular potential (Case N; centre panel) the Figs. 3 to 6 is the isotropic coordinate radius of the ADM-CFC profile deviates clearly from both the corresponding solution of metric used in the relativistic version of the CoCoNuT code A.Mareketal.:Animprovedeffective potential for supernova simulations 281

16 6 4 Newtonian with modified TOV potential A ]

-3 3 14 AB-GR ] ]

V-R 3 3 g cm V-A 5 14 2 ρ cm cm nuc V-B / / g

[10 V-F

[g 12

ρ 14

1 c ρ [10

0 4 c ρ log 4 Newtonian with modified TOV potential B 10

] -3 3 8 3

g cm -200 -100 0 0 100 200 300 14 2 ρ nuc t − tb [ms] [10 ρ 1 Fig. 7. Time evolution of the central density ρc for Model AB-GR (bold solid line), V-A (solid line), V-B (dashed-dotted line), 0 0 5 10 15 V-F (dashed line), and V-R (dotted line). The left panel shows the col- r [km] lapse phase (note that here all models with the Vertex code lie on top of each other), while the right panel shows the post-bounce evolution. Fig. 5. Comparison of radial profiles of the density ρ from the hydro- Note the different axis scales in both panels. dynamic evolution of Model AS at late times (solid line) and from a solution of the TOV structure equations with identical central density (dashed line). In the Newtonian simulations with either modification A relativistic coordinate time (or proper time) when comparing (top panel)orB(bottom panel) of the TOV potential, the two curves the profiles at the same coordinate times, as the compact rem- match well and also agree with the result from the relativistic simula- nants are in equilibrium at “late” times (about 60 ms after core tion (Case GR; dashed-dotted line). bounce). 0.14 Newtonian with modified TOV potential A 4.2. Supernova core collapse with microphysical 0.12 equation of state and neutrino transport ] 2

c ∋

[ Next we explore the moderately relativistic regime of stellar 0.10 core collapse with the microphysical EoS of Lattimer & Swesty (1991) and neutrino transport. We simulated the collapse and 0.08 the post-bounce evolution of the progenitor model s15s7b2 Newtonian with modified TOV potential B with the Vertex code as detailed in Sect. 3.2. The calculations were performed using the TOV potential given in Eq. (2) 0.12 ertex ] (Model V-R, which is identical with the V calculation 2

c ∋ of Model G15 in Liebendörfer et al. 2005), and we also tested [ 0.10 the modifications A, B, and F of the TOV potential (Models V-A, V-B, and V-F, respectively). For comparison, we refer in the following discussion also to the calculation of Liebendörfer 0.08 0 5 10 15 et al. (2005) with the fully relativistic Agile-BoltzTran code r [km] (Case GR, Model AB-GR). Fig. 6. Same as Fig. 5, but for radial profiles of the specific internal Figure 7 shows the central density as a function of time energy . The step-like shape of the profile arises from contributions for the collapse (left panel) and for the subsequent post-bounce of the thermal energy th in various strengths. phase (right panel). The “central” density is the density value at the centre of the innermost grid zone of the AB-GR simulation. Because of a different numerical resolution it was necessary to (for details, see Dimmelmeier et al. 2002a). In contrast to this, interpolate the Vertex results to this radial position. During the TOV Eqs. (A.1), (A.2) use the standard Schwarzschild-like the collapse only minor differences between the relativistic cal- coordinate radius, which is equivalent to the circumferential culation (bold solid line) and the calculations with the Vertex radius. Thus, a coordinate transformation from the standard code are visible. Note that the trajectories from the Vertex Schwarzschild-like radial coordinate to the isotropic radial co- code, with the modifications A, B, and F of the TOV potential ordinate (as specified in Appendix B) is applied to all profiles of as well as with the TOV potential (case R), lie on top of each quantities obtained with the TOV solution or Newtonian sim- other. ulations in those figures. Otherwise, the difference in the two We can thus infer that the differences between the modifi- radial coordinates of up to ∼20% would lead to a noticeable cations A, B, and F of the TOV potential are unimportant dur- distortion of the radial profiles. We also point out that we ne- ing the collapse phase. Furthermore, we can conclude from the glect any influence of the difference between Newtonian and good agreement of the general relativistic calculation and the 282 A.Mareketal.:Animprovedeffective potential for supernova simulations

200 eﬀects but is a consequence of a diﬀerent numerical tracking of the time evolution of interfaces between composition layers 150 in the collapsing stellar core (for more details about the numer- ics and a discussion of the involved physics, see Liebendörfer et al. 2005). It is therefore irrelevant for our present compari- 100 son of approximations to general relativity. A good choice for [km]

s ff r the e ective relativistic potential (like Case A) should just en- AB-GR V-R sure that the corresponding shock trajectory converges again 50 V-A with the relativistic result (Case GR) after the transient period V-B of shock expansion. V-F 0 The time evolution of the central density or the shock po- 0 50 100 150 200 250 300 sition, however, is not the only relevant criterion for assessing t − t [ms] b the quality of approximations to general relativity, as already Fig. 8. Time evolution of the shock position rs for simulations with the discussed in Sect. 4.1. A good approximation does not only re- Vertex code using various modifications of the TOV potential com- quire good agreement for particular time-dependent quantities pared to the general relativistic result from the Agile-BoltzTran (like, e.g., the central density), but also requires that the radial code (Model AB-GR, thick solid line). structure of the models reproduces the relativistic case as well as possible at any time. Vertex calculations that the TOV potential works well dur- In the left panels of Fig. 9 we show such profiles of the den- ing the collapse phase. However, after core bounce this poten- sity and velocity (top panel) and of the entropy and electron tial overestimates the compactness of the forming neutron star fraction Y , i.e., the electron-to-baryon ratio (bottom panel), (just as in the purely hydrodynamic simulations in Sect. 4.1), e for Models AB-GR and V-A at a time of 250 ms after bounce, and therefore the density trajectories of Model AB-GR and when the discrepancy between relativistic and approximative Model V-R diverge (see Fig. 7). At 250 ms after the shock for- treatment was found to be largest in Liebendörfer et al. (2005). mation the central density in Model V-R is about 20% higher Fig. 9 can be directly compared with Fig. 12 in the latter ref- than the one in the relativistic calculation. At this time the mod- erence. Obviously Model V-A fits the density profile of the ifications A and B of the TOV potential give a central density general relativistic calculation (Model AB-GR) extremely well only about 2% higher than Model AB-GR, and the absolute dif- at all radii. Furthermore, both the velocity ahead of the shock ference stays practically constant during the entire post-bounce front and behind it are in extremely good agreement between evolution. This implies that both modifications yield very good the two models (the differences at the shock jump have prob- quantitative agreement with the general relativistic treatment. ably a numerical reason associated with the different handling In contrast, in Model V-F the central density after bounce is of shock discontinuities in both codes). It is an important re- lower than the relativistic result of Model AB-GR. This indi- sult that with this modification A of the TOV potential one is cates a strong underestimation of the depth of the gravitational able to approximate the kinematics of the relativistic run with potential in Case F, where Γ=1 in the integrand of Eq. (2). astonishingly good quality in a Newtonian calculation (at least Since the central densities suggest that differences between in supernova simulations when the velocities do not become a fully relativistic calculation and Newtonian simulations with highly relativistic). In contrast, in Vertex runs with the origi- effective relativistic potential become significant only after nal TOV potential (Case R) the pre-shock velocities were found shock formation (see also Liebendörfer et al. 2005), we discuss to be significantly too large (Liebendörfer et al. 2005), which is the implications of our potential modifications in the following due to the overestimated strength of gravity in the far field limit only during the post-bounce evolution. as compared to the relativistic calculations (see the discussion Figure 8 shows the shock positions as functions of time. in Sect. 2.3). Both Case A (thin solid line) and Case B (dashed-dotted line) reveal the desirable trend of a closer match with the general rel- Also the entropy and Ye profiles (bottom left panel of Fig. 9) ativistic calculation (thick solid line) than seen for Model V-R, reveal a similarly excellent agreement between Models V-A which gives a shock radius that is too small, and Model V-F, and AB-GR. The minor entropy differences ahead of the shock where the shock is too far out at all times. In particular, are associated with a slightly different description of the mi- Model V-A reveals excellent agreement with Model AB-GR. crophysics (nuclear burning and equation of state) in the infall ff The only major di erence is visible between 170 ms and about region (for details we refer to Liebendörfer et al. 2005). 230 ms after shock formation when the shock transiently expands in the Vertex calculation. This behaviour is generic for Not only the radial structure of the forming neutron star in the Vertex results and independent of the choice of the grav- all relevant quantities is well reproduced, but also the neutrino itational potential. In the Agile-BoltzTran run the transient transport results of the relativistic calculation and of the ap- shock expansion is much less pronounced and also a bit de- proximative description of Case A are in nearly perfect agree- layed relative to the Vertex feature (it is visible as a decelera- ment. Corresponding radial profiles of the luminosities and tion of the shock retraction between about 200 ms and 250 ms). root mean square energies – both as defined in Sect. 4 of This difference, however, is not caused by general relativistic Liebendörfer et al. (2005) – for electron neutrinos, νe, electron A.Mareketal.:Animprovedeffective potential for supernova simulations 283 60 t = 250 ms t = 250 ms 1014 0 νe ρ 50 ν¯e νx s] s] v 12 ] 40 / 3 -2 10 / cm cm erg / 9 30 51 [g

[10 10 ρ

-4 10 [10 v

L 20

-6 108 10 20 0.6 0 ν 0.5 e ν¯e 15 100 νx Ye 0.4 s AB-GR e [MeV] baryon] 10 0.3 Y V-A / B k RMS [

s ν

0.2 5 AB-GR 0.1 V-A 10 0 0.0 0 50 100 150 200 0 50 100 150 200 r [km] r [km]

Fig. 9. Left: radial profiles of the velocity v (dashed lines) and density ρ (solid lines) for Model V-A (thin) and the relativistic Model AB-GR (bold) at a time of 250 ms after shock formation (top panel), as well as radial profiles of the entropy s (dashed lines) and of the electron fraction Ye (solid lines) for the same models and the same time (bottom panel). As in Liebendörfer et al. (2005), r is the circumferential radius in case of the relativistic results. Note the different vertical axes on both sides of the two panels. Right: radial profiles of the luminosities L of electron neutrinos (solid lines), electron anti-neutrinos (dotted lines), and heavy-lepton neutrinos (dashed lines) for Models V-A (thin) and AB-GR (bold) at a time of 250 ms after shock formation (top panel), as well as radial profiles of the root mean square energies ν rms for the number densities of νe,¯νe, and heavy-lepton neutrinos for Models V-A and AB-GR (bottom panel). The labeling is the same as in the panel above, and all neutrino quantities are given for a comoving observer.

1 antineutrinos,ν ¯e, and heavy-lepton neutrinos , νx,aredis- Liebendörfer et al. 2005). A detailed analysis reveals that both played in the right panels of Fig. 9. The results for Models V-A codes produce internally consistent results, conserving to good and AB-GR for all neutrino flavours share their characteristic precision the luminosity through the shock for an observer at features, and in particular agree in the radial positions where rest and showing the expected and physically correct behav- the different luminosities start to rise. While the νx luminosi- ior in the limit of large radii. Most of the observed difference ties are nearly indistinguishable below the shock, the jump at (which has no mentionable significance for supernova mod- the shock is slightly higher for the Vertex run and reflects the eling) could be traced back to the fact that Vertex achieves larger effects due to observer motion, e.g., Doppler blueshift only order (v/c) accuracy, whereas Agile-BoltzTran pro- and angular aberration, for an observer comoving with the duces the full relativistic result including higher orders in (v/c). rapidly infalling stellar fluid ahead of the shock. The offset be- Corresponding effects become noticeable when v/c ∼> 0.1. The tween results of Models V-A and AB-GR decreases at larger mean neutrino energies are hardly affected by this difference radii where the infall velocities are lower. This discrepancy was (Fig. 9, bottom right panel). In case of the νe andν ¯e lumi- not discovered by Liebendörfer et al. (2005), because there the nosities the Vertex run yields roughly 10% lower values out- agreement of the radial structure for both investigated models side of the corresponding neutrino spheres (i.e., between about was generally found to be poorer than in the present work. 50 km and 90 km), but values much closer to those from the General relativistic effects are unlikely as an explanation, Agile-BoltzTran calculation ahead of the shock. Since the because they are very small around the shock (see Fig. 13 in neutrinospheric emission of νe andν ¯e is strongly affected by the mass accretion rate of the nascent neutron star and the cor- 1 Since the transport of muon and tau neutrinos and antineutrinos responding accretion luminosity (which both seem to have the ff di ers only in minor details we treat all heavy-lepton neutrinos iden- tendency of being slightly higher in Model AB-GR), we refrain tically in the Vertex simulations. 284 A.Mareketal.:Animprovedeffective potential for supernova simulations ν 50 30 x t = 250 ms

s] 14 40 s] / 25 / 0 10 20 g erg

30 er ρ 51 15 51 20 s]

[10 v 12

10 ] [10 / -2 10 3 L 5 10 L cm 0 0 cm / 30 ν 9 ¯ [g e 60 [10 s] s] 10 ρ

/ 25 / v -4 10 20 g erg 40 er 51 15 51

[10 10 20 [10 -6 108 L 5 L 0 0 20 80 4 νe 100 AB-GR Lν V-R e 80 15 60 3 V-A s

V-B s] s] s] / / /

60 g erg erg er

baryon] 10 40 53 51 51

2 / B k [10 [10 [ 40 [10 s L L L

1 5 20 20 V-R V-A V-B 0 0 0 0 0 5 1015 0 100 200 300 0 50 100 150 200 − t tb [ms] r [km] Fig. 10. Luminosities as functions of post-bounce time for different Fig. 11. Radial profiles of the velocity v (dashed lines) and den- cases computed with the Vertex code and for Model AB-GR. The sity ρ (solid lines) for Models V-A (bold lines), V-B (thin), and V-R top panel shows the results for heavy-lepton neutrinos, the centre (medium) at a time of 250 ms after shock formation (top panel). panel those for the electron antineutrinos, and the bottom panel the Radial profiles of the entropy s (dashed lines) and of the electron neu- results for electron neutrinos. The panels on the left magnify the early trino luminosity Lνe (solid lines) for the same models and the same post-bounce phase. All luminosities are given for an observer comov- time (bottom panel). The luminosity is given for an observer comoving with the stellar fluid at a radius of 500 km. Note the different scales ing with the stellar fluid. of the vertical axes.

the overestimated compactness of the proto-neutron star in from ascribing the different magnitude of the νe andν ¯e lumi- Model V-R, which causes the faster contraction of the stalled nosities only to the treatment of relativistic effects. Although shock after maximal expansion (Fig. 8), also leads to higher such a connection cannot be excluded, the luminosity differ- neutrino luminosities during the accretion phase. Consistent ences might (partly) also be a consequence of the different ac- with the shock trajectories, Model V-A yields the closest match cretion histories in Models AB-GR and V-A, which manifest with the general relativistic run of Model AB-GR also for the themselves in the shock trajectories (Fig. 8) and are attributable neutrino luminosities. It is very satisfactory that the results to the different handling of the microphysics and computational (shock radius rs as well as the neutrino luminosities L) from grid in the infall layer (see above and Liebendörfer et al. 2005). both simulations reveal convergence at later times when the pe- This interpretation seems to be supported by the time evolu- riod of massive post-bounce accretion comes to an end. tion of the neutrino luminosities plotted in Fig. 10. The accre- In Fig. 11 we present the radial structure at 250 ms af- tionbumpintheνe andν ¯e luminosities which follows after the ter bounce for the Vertex simulations with the modifica- prompt νe burst is stretched in time in case of Model AB-GR, tions A and B of the TOV potential, compared to the results indicating the delay of mass infall at higher rates relative to with the TOV potential (Case R) which was already discussed all Vertex simulations. Note that the neutrino emission reacts in Liebendörfer et al. (2005). Note that because of the excel- with a time lag of some 10 ms (corresponding to the cooling lent agreement seen in Fig. 9, Model V-A (Case A) can also timescale of the accretion layer on the neutron star) to varia- be considered as a representation of the fully relativistic run tions of the mass accretion rate. of Model AB-GR. Models V-A and V-B show results of sim- Moreover, Fig. 10 shows that our variations of the effective ilar quality. The little offset of the shock position (which is relativistic potential in the Vertex models have little influence causally linked to the differences in all profiles) might sug- on the prompt burst of νe at shock breakout. But subsequently gest that Model V-B is slightly inferior to Model V-A in A.Mareketal.:Animprovedeffective potential for supernova simulations 285 approximating relativity. This conclusion could also be drawn from the post-bounce luminosities in Fig. 10. However, cau- 4 tion seems to be advisable with such an interpretation, being aware of the uncertainties in the accretion phase and infall 3 layer discussed above, and in view of the fact that the cen- M ] r · tral densities (Fig. 7) and radial density profiles (Fig. 11) agree O M

well. Moreover, the quality of the agreement at “very late” [ g 2 times cannot be judged, because no information is available M for the behaviour of Model AB-GR after 250 ms post bounce, • • a time when the settling of the shock radius and luminosities ρ ρ 1 c f c i to their post-accretion levels seems not yet over in this model (Figs. 8, 10). The TOV potential of Case R clearly produces too large infall velocities ahead of the shock (and therefore does 0 not agree well with the kinematics of the relativistic calcula- 1 10 100 ρ -3 tion), overestimates the compactness of the forming neutron c [g cm ] star, and thus produces too high neutrino luminosities during the simulated period of evolution (for a detailed discussion, see 50 Liebendörfer et al. 2005). Cases A and B clearly perform bet- ρ c i ter and must be considered as signiﬁcant improvements for use 40 ﬀ

in Newtonian simulations with an e ective relativistic potential ] as approximations to fully relativistic calculations. -3 30 g cm 14 4.3. Neutron stars in equilibrium – Migration test

[10 20

c ρ

ρ c f The supernova core collapse simulations presented in Sect. 4.1 and 4.2 are limited to a maximum core compactness 10 2Mcore/Rcore ∼ 0.2. In order to extend the assessment of the quality of a Newtonian simulation with an effective relativistic 0 potential to a more strongly relativistic regime, we construct 01234 spherical equilibrium models of neutron stars using a simple t [ms] polytropic EoS, P = Kργ, with γ = 2andK = 1.455 × 105 Fig. 12. Mass-density diagram for spherical equilibrium neutron star (in cgs units). The TOV equations are solved on a very fine grid 5 models (top panel) and time evolution of the central density for the with 10 equidistantly spaced radial grid points using a second- migration test from the unstable to the stable branch (bottom panel). order accurate Runge–Kutta integration scheme. By varying In both graphs the behaviour of the Newtonian models computed with ρ = . × 14 −3 the value of the central density from c 1 0 10 gcm the modification A (dotted line) and B (long-dashed line) of the TOV − to 100.0 × 1014 gcm 3, we cover the entire range from weak to potential is close to that in the relativistic case (Case GR; solid line). strong relativistic gravity. The density variation corresponds to The Newtonian equivalent of the gravitational mass Mg (dashed line) compactness parameters ranging from 2Mg/R ∼ 0.02 to 0.5, as well as the rest mass Mr (dashed-dotted line) calculated using a where Mg is the total gravitational mass of the neutron star and regular Newtonian potential (Case N) deviate strongly from the rela- ρ R its radius. We again point out that in spherical symmetry the tivistic solution (top panel). While c oscillates around its initial value ρ assumption of CFC constitutes no approximation. ci for Case N (dashed line), in Case R (dashed-dotted line) the neutron star disperses to a state with ρc ρcf (bottom panel). The mass-density curve of the corresponding TOV neutron star models (Fig. 12, top panel) exhibits the well-known form consisting of a stable branch at low densities and an unstable relativistic gravitational mass is introduced in Newtonian grav- branch at high densities, where the gravitational mass Mg in- ity, Mg ≡ Mr − Mb,whereMb ≡|Eb| is the mass associated ρ creases or decreases with c, respectively. The two branches with the binding energy of the self-gravitating object, we find meet at the maximum mass. Configurations on the unstable negative values at large densities, where the binding energy be- branch are unstable to small perturbations and either collapse to comes so large that Mb > Mr. a black hole or migrate to a configuration on the stable branch On the other hand, when using the modifications A and B with the same gravitational mass but a lower central density of the TOV potential in the TOV equation for the neutron star ρ <ρ cf ci (as indicated by the arrow for an exemplary configu- model sequence, the corresponding curves for the TOV gravi- ration in the top panel of Fig. 12). tational mass Mg in Fig. 12 agree very well with the relativis- While the gravitational mass Mg and the rest mass Mr are tic curve (Case GR) even in the high density regime, and they limited in relativistic gravity (Case GR), Mg Mr 1.6 M, show the same qualitative behaviour with an increasing and regular Newtonian gravity (Case N) allows for solutions with a decreasing branch. The respective mass maxima are located arbitrarily high rest mass Mr, which are all stable against close to the relativistic one for both modifications of the TOV (small) perturbations. If an (unphysical) equivalent to the potential. 286 A.Mareketal.:Animprovedeffective potential for supernova simulations

Note that for constructing equilibrium configurations us- initial equilibrium state with ρc(t) ρci. On the other hand, ing the TOV potential (Case R) is identical to solving the fully using the TOV potential (Case R) leads to a rapid dispersion relativistic TOV equations (Case GR), and thus the results are (without ring-down oscillations) of the neutron star with a final identical. Therefore, this test is useless for predicting the qual- central density ρc ρcf, which is in strong qualitative dis- ity of the TOV potential in the context of a dynamic evolution. agreement with the behaviour in relativistic gravity (Case GR). This is different when a neutron star model from the unsta- We thus conclude that in the demanding test case of a migrat- ble branch is allowed to evolve in time using a dynamic code. ing spherical neutron star model, both the regular Newtonian Then the (small but nonzero) finite difference discretisation er- potential (Case N) and the TOV potential (Case R) fail, while rors of the numerical scheme act as perturbations of the un- the modifications A and B of the TOV potential reproduce stable equilibrium and excite small oscillations. Depending on the relativistic result very well, both qualitatively and also the numerical algorithms used in the code and the resolution quantitatively. of the grid, the neutron star either contracts to higher densities (and ultimately to a black hole for a simple polytropic EoS) 5. Multi-dimensional simulations with rotation or approaches a new equilibrium state on the stable branch af- For the multi-dimensional calculations we only use the ter a strongly dynamical, nonlinear evolution phase. This neu- CoCoNuT code, since parameter studies with MudBath (the tron star migration scenario is a standard test of the relativistic two-dimensional version of Vertex) are computationally too regime with strongly dynamical evolution for numerical hydro- expensive when neutrino transport is included. Moreover, cur- dynamic codes with relativistic gravity (see, e.g., Font et al. rently there exists no other fully relativistic two-dimensional 2002; Baiotti et al. 2005). code including neutrino transport to compare our results with. The time evolution of the central density is depicted in The following discussion hence focuses on purely hydrody- the bottom panel of Fig. 12 for such a migration to the sta- namic simulations using the simple matter model described in ble branch, where the central density ρc drops from ρci = Sect. 3.1. 14 −3 14 −3 49.3 × 10 gcm to ρcf = 8.8 × 10 gcm . Shown is the Motivated by the promising results in spherical symmetry, evolution for a relativistic simulation (Case GR) as well as for we now apply the modified effective relativistic potentials ac- Newtonian simulations utilizing a regular Newtonian potential cording to Eq. (7) to simulations of rotational core collapse in (Case N), the TOV potential (Case R), and its modifications A axisymmetry. The initial models of the pre-collapse core are and B. For these simulations we again use the CoCoNuT described by the same EoS parameters as the ones in Sect. 4.1 10 −3 code, whose implementation of the hybrid EoS, Eqs. (12), (13) with ρci = 10 gcm , but now rotate at various rotation allows one to suppress the contribution of the thermal pres- rates with different degrees of differential rotation. To construct sure Pth, in which case the hybrid EoS reduces to a polytropic these rotating equilibrium configurations, we use Hachisu’s one. self-consistent field method (see Komatsu et al. 1989). The ini- As expected and as observed in simulations with similar tial pressure reduction to trigger the collapse and the stiffening relativistic codes, in relativistic gravity (Case GR) the neutron of the EoS follows the prescription given in Sect. 4.1. star initially expands rapidly and its central density decreases For these multi-dimensional tests we choose three rota- strongly. It undershoots the stable equilibrium at ρcf and expe- tional core collapse models as representative cases. Their evo- riences several ring-down oscillations of decreasing amplitude lution is influenced by the effects of rotation at various de- until ρc approaches ρcf at late times. The modifications A and B grees (for details about the specifications of these models, see of the TOV potential both have initial (final) states on the un- Dimmelmeier et al. 2002a). The core of Model A1B3G1 is an stable (stable) branch of the mass-density curve which are very almost uniform rotator with a moderate rotation rate and col- close to the relativistic one (Case GR). Therefore the evolution lapses nearly spherically, forming a compact remnant imme- of ρc (starting again with the same central density ρci) is similar diately after core bounce. Model A4B1G2 is a very differen- to the relativistic case, as shown in Fig. 12. The disagreement tially rotating model, which develops considerable deviations in oscillation amplitudes and periods can be attributed to the from spherical symmetry during its evolution, but otherwise differences in the Newtonian and relativistic kinematics. When shares the qualitative bounce behavior of the previous model comparing the results of migration tests using various relativis- A1B3G1. Finally, in the multiple bounce model A2B4G1 the tic hydrodynamic codes (Stergioulas et al. 2003), we observe increase of the centrifugal forces during the contraction phase that even small variations in the numerical approach, e.g., a causes a series of subsequent multiple bounces at subnuclear different coordinate choice or grid setup, or differences in the densities. The time evolution of the central density for these treatment of the artificial atmosphere, can have a strong impact three models computed relativistically (Case GR) is shown on the detailed evolution of ρc from the initial to the final equi- with solid lines in the top, centre, and bottom panels of Fig. 13, librium state. Consequently, we consider the results obtained respectively. with the modifications A or B of the TOV potential in an other- In a Newtonian simulation with regular Newtonian gravity wise Newtonian formulation to be a very good approximation (Case N) all three models exhibit several consecutive bounces of the relativistic one (Case GR). (at subnuclear densities for Models A1B3G1 and A2B4G1, Using regular Newtonian gravity (Case N), however, yields and just above nuclear matter densities for Model A4B1G2), a totally different behaviour, because there exist no unstable with the central density being considerably lower during the configurations in that case. Thus, discretisation errors cannot entire evolution (Fig. 13). Such a qualitative change of the col- trigger a migration, and the star simply oscillates around its lapse dynamics between a Newtonian and a general relativistic A.Mareketal.:Animprovedeffective potential for supernova simulations 287

in Fig. 2. Case R yields a central density noticeably higher than 4 A1B3G1 the one in the relativistic simulation (Case GR). For Case A the result agrees very well with the relativistic one, while for ]

-3 3 Case B the central density is too small. As the inﬂuence of rotation increases, the ordering of the g cm curves for Cases R, A, and B (from higher to lower ρc) remains 14 2 ρ nuc unaltered. However, all curves are shifted to lower densities [10 c

ρ (Fig. 13). For Model A4B1G2 the result from the Newtonian 1 simulation with the TOV potential (Case R) matches the relativistic result (Case GR) best, which is a coincidence associated with the particular rotational state of this model. For 0 0102030Model A2B4G1, where the inﬂuence of rotation is stronger, − ﬀ t tb [ms] the results obtained for Cases A, B, and R di er considerably from the relativistic results. A behaviour consistent with that of these three representa- 4 A4B1G2 tive models is observed for other rotational core collapse models as well. Thus, in rotational core collapse, even a Newtonian ]

-3 3 simulation with the TOV potential (Case R) yields lower central densities than the corresponding relativistic simulation g cm (Case GR) beyond a certain rotation rate (which depends both 14 2 ρ nuc on the initial model and on the EoS parameters). [10 c

ρ We therefore conclude that for rotational core collapse the 1 use of neither the modifications A and B of the TOV potential nor the (stronger) TOV potential itself (Case R) can closely reproduce the collapse behaviour of a relativistic code for all de- 0 -5 0 5 10 grees of rotation. Hence, in a Newtonian code either the effec- − ff t tb [ms] tive relativistic potential has to include the e ect of rotation, or the collapse kinematics (and thus the hydrodynamic equations) have to be modified. However, in the latter case the advantage A2B4G1 0.6 of a Newtonian approach over a fully relativistic formulation in terms of simplicity and computational speed may be lost. ]

-3 On the other hand, we ﬁnd that the qualitative dynami- 0.4 cal behaviour of relativistic rotational core collapse models g cm

14 (Case GR), particularly their collapse type (regular collapse, multiple bounce collapse, or rapid collapse as introduced by [10 c ρ 0.2 Zwerger & Müller 1997) is correctly captured in a Newtonian simulation for a wide range of parameter space, if the TOV potential (Case R) or its modified versions (Cases A to D) are 0.0 used. Moreover, the corresponding results agree much better -200 20406080with the relativistic result than that obtained from a Newtonian − t tb [ms] simulation with regular Newtonian potential (Case N). Hence, the usage of the effective relativistic potentials is an improve- Fig. 13. Time evolution of the central density ρc for the rotating core collapse models A1B3G1 (top panel), A4B1G2 (centre panel), and ment of Newtonian simulations also in the case of rotational A2B4G1 (bottom panel), respectively. With increasing influence of core collapse. rotation on the collapse dynamics, the results of the Newtonian simulations with the modifications A (dotted line) and B (long-dashed 6. Summary and conclusions line) of the TOV potential and eventually also with the TOV potential (Case R; dashed-dotted line) move away from the relativistic re- We investigated different modifications of the TOV potential sult (Case GR; solid line). The central density obtained with regular used as effective relativistic potentials in Newtonian hydro- Newtonian gravity (Case N; dashed line) is lowest for all three models. dynamics with the aim to test their quality of approximating relativistic effects in stellar core collapse and supernova simulations. This work was motivated by a recent compar- simulation of the same model was also observed in the compre- ison of the neutrino radiation-hydrodynamics codes Agile- hensive parameter study of rotational core collapse performed BoltzTran and Vertex used by the Oak Ridge-Basel collab- by Dimmelmeier et al. (2002b). oration and by the Garching group, respectively (Liebendörfer When applying the TOV potential (Case R) or its modifi- et al. 2005). While the former code is a fully relativistic cations A and B to Model A1B3G1 (top panel of Fig. 13) the implementation of neutrino transport and hydrodynamics in outcome is similar to the situation in spherical symmetry shown spherical symmetry, the latter code employs an approximative 288 A.Mareketal.:Animprovedeffective potential for supernova simulations relativistic description by treating the self-gravity of the stel- the Vertex code were already presented in Liebendörfer et al. lar fluid with an effective relativistic potential tailored from a (2005). Good agreement between the relativistic and approxi- comparison of the Newtonian and relativistic equations of mo- mative treatments was seen in the transport quantities as well. tion. Otherwise Vertex solves the Newtonian equations of hy- The Vertex code performs well despite the fact that the ap- drodynamics, ignoring the effects of relativistic kinematics and proximative description takes into account only relativistic red- space-time curvature (Rampp & Janka 2002). The comparison shift in the transport but – for reasons of consistency with the by Liebendörfer et al. (2005) shows very good agreement of Newtonian hydrodynamics code – ignores the difference be- the results from both codes for the phases of stellar core col- tween proper radius and coordinate radius. Remaining (minor) lapse, bounce and prompt shock propagation. However, it also quantitative differences (below about 10% for neutrino data revealed a moderate but gradually increasing overestimation of and even smaller for hydrodynamic quantities) are probably the compactness of the nascent neutron star during the subse- only partly associated with relativistic effects. We suspect that quent accretion phase, accompanied by an overestimation of a significant contribution to these differences originates from the luminosities and mean energies of the emitted neutrinos. small (but unavoidable because linked to the specifics of the These systematic differences suggested that an improve- employed numerical methods) discrepancies in the infall layer ff ahead of the shock as described by the Vertex and Agile- ment might be possible by changing the e ective relativistic oltz ran potential such that the strength of the gravitational field is re- B T codes. duced relative to the TOV potential. To this end we consid- The results discussed in this paper therefore demonstrate ered a variety of modifications (Cases A–F; see Sect. 2.2) of that a very good, simple, and computationally efficient approx- the TOV potential (which defines our reference Case R), and imation to a relativistic treatment of stellar core collapse and compared corresponding hydrodynamic as well as hydrostatic neutron star formation in spherical symmetry can be achieved solutions with regular Newtonian (Case N) and fully relativis- by modeling the self-gravity of the stellar plasma with an ef- tic calculations (Case GR). While some modifications did not fective relativistic potential based on the TOV potential as sug- provide any significant advantage over the TOV potential, sev- gested in this work. Two-dimensional axisymmetric collapse eral options for the potential were identified which allow for a calculations (using a simple equation of state and ignoring neu- much better reproduction of relativistic results with Newtonian trino transport) for rotating stellar cores with a wide range of hydrodynamics. In particular, reducing the TOV mass, Eq. (4), conditions showed that even in this case the TOV potential and that appears in the expression of the TOV potential, turned out its modifications reproduce the characteristics of the relativistic to be very successful in this respect. collapse dynamics quantitatively well for not too rapid rotation. For example, in Case A we introduced an additional met- There is still good qualitative agreement when the rotation be- ric factor Γ < 1, Eq. (5), in the integrand of the TOV mass, comes fast, in contrast to Newtonian simulations with regular and in Case B we ignored the internal energy (and all terms Newtonian potential (Case N), which mostly fail even quali- ff depending on neutrinos) in the TOV mass. In both cases not tatively. Multi-dimensional simulations with an e ective rela- only characteristic model parameters like the central density tivistic potential (which in general does not satisfy the Poisson and shock radius were in excellent agreement with relativis- equation), however, fulfill strict momentum conservation only tic simulations during all stages of the evolution including the for special cases of symmetry (for details, see Sect. 3.3). long-term accretion phase of the collapsing core after bounce. Although our Cases A and B yield results of similar quality Also the radial profiles of the hydrostatic central core as ob- for all the test problems considered here and neither is pre- tained in the hydrodynamic simulations were found to be very ferred when a simple equation of state is used, the situation is good approximations of the profiles from the fully relativistic more in favour of Case A when microphysics is taken into ac- treatment. Consequently, they are nearly consistent with solu- count in the equation of state. When particles can annihilate or tions of the TOV equations for equal central density. Migration nuclear interactions take place, rest mass energy is converted tests of extremely relativistic, ultradense, hydrostatic config- into internal energy and vice versa. In such a situation baryon urations from the unstable to the stable branch of the mass- number and lepton number are conserved, but not the total rest density relation confirmed the good qualitative and quantitative mass of the particles, and only the total (“relativistic”) energy performance of the approximation even for such a demanding density, ρ + e, is a well defined quantity, but not the individual test problem with a strongly dynamic transition from the initial energies. Case B therefore becomes ambiguous (see Sect. 2.2) to the final state. This suggests that the use of a modified TOV while Case A does not suffer from such problems. We there- potential in an otherwise Newtonian hydrodynamics code also fore are tempted to recommend using Case A as an effective ensures a good reproduction of relativistic kinematics. relativistic potential in Newtonian hydrodynamics codes. The In addition to these purely hydrodynamic simulations with extra factor Γ < 1 in the integrand of the TOV mass, Eq. (4), the CoCoNuT code, we performed runs including neutrino reduces the mass integral below the mass equivalent of the to- transport and compared them to relativistic calculations with tal (rest mass plus internal) energy. Its introduction may be the Agile-BoltzTran code of the Oak Ridge-Basel collab- justified by heuristic arguments and consistency considerations oration (cf. the simulations for Model G15 in Liebendörfer when the relativistic concept of the gravitating mass is applied et al. 2005). Also in this application, the modifications A and B in a Newtonian description of the dynamics, which does not of the TOV potential produced a major improvement relative distinguish between proper volumes and coordinate volumes to the original TOV potential (Case R), for which results with (see Sect. 2.3). A.Mareketal.:Animprovedeffective potential for supernova simulations 289

Acknowledgements. We thank M.-A. Aloy and M. Obergaulinger for Isenberg, J. A. 1978, University of Maryland Preprint, unpublished helpful discussions and M. Rampp for his support. We are particu- Janka, H.-T., Zwerger, T., & Mönchmeyer, R. 1993, A&A, 268, larly grateful to the referee, M. Liebendörfer, for his interest in our 360 work and his knowledgeable comments, which helped us to improve Kippenhahn, R., & Weigert, A. 1990, Stellar Structure and Evolution, our manuscript. We also thank him for providing us with data of Astronomy and Astrophysics Library (Berlin, Germany: Springer) his general relativistic run. This work was supported by the German Komatsu, H., Eriguchi, Y., & Hachisu, I. 1989, MNRAS, 237, 355 Research Foundation DFG (SFB/Transregio 7 “Gravitationswellen- Lattimer, J. M., & Swesty, F. D. 1991, Nucl. Phys. A, 535, 331 astronomie” and SFB 375 “Astroteilchenphysik”). The simulations Liebendörfer, M., Messer, O. E. B., Mezzacappa, A., et al. 2004, were performed at the Max-Planck-Institut für Astrophysik and the ApJS, 150, 263 RZG Rechenzentrum in Garching. Liebendörfer, M., Mezzacappa, A., & Thielemann, F.-K. 2001, Phys. Rev. D, 63, 104003 Liebendörfer, M., Rampp, M., Janka, H.-T., & Mezzacappa, A. 2005, References ApJ, 620, 840 Baiotti, L., Hawke, I., Montero, P. J., et al. 2005, Phys. Rev. D, 71, Liebendörfer, M., Rosswog, S., & Thielemann, F.-K. 2002, ApJS, 141, 024035 229 Baron, E., Myra, E. S., Cooperstein, J., & van den Horn, L. J. 1989, Müller, E., & Steinmetz, M. 1995, Computer Phys. Commun., 89, 45 ApJ, 339, 978 Ott, C. D., Zink, B., Dimmelmeier, H., et al. 2005, in preparation Cerdá-Durán, P., Faye, G., Dimmelmeier, H., et al. 2005, A&A, 439, Rampp, M., & Janka, H.-T. 2002, A&A, 396, 361 1033 Shapiro, S. L., & Teukolsky, S. A. 1983, Black holes, white dwarfs, Cook, G. B., Shapiro, S. L., & Teukolsky, S. A. 1996, Phys. Rev. D, and neutron stars: The physics of compact objects (New York, NY, 53, 5533 USA: Wiley) Dimmelmeier, H., Font, J. A., & Müller, E. 2002a, A&A, 388, 917 Shibata, M., & Sekiguchi, Y. 2004, Phys. Rev. D, 69, 084024 Dimmelmeier, H., Font, J. A., & Müller, E. 2002b, A&A, 393, 523 Shu, F. H. 1992, The Physics of Astrophysics, Vol. II, Gas Dynamics Dimmelmeier, H., Novak, J., Font, J. A., Ibáñez, J. M., & Müller, E. (Mill Valley, CA, USA: University Science Books) 2005, Phys. Rev. D, 71, 064023 Stergioulas, N., Zink, B., & Font, J. A. 2003, private communication Font, J. A., Goodale, T., Iyer, S., et al. 2002, Phys. Rev. D, 65, 084024 van Riper, K. A. 1979, ApJ, 232, 558 Fryxell, B., Müller, E., & Arnett, D. 1990, in Proceedings of the 5th Wilson, J. R., Mathews, G. J., & Marronetti, P. 1996, Phys. Rev. D, Workshop on Nuclear Astrophysics, Ringberg Castle, 30 January– 54, 1317 4 February, 1989, ed. W. Hillebrandt, & E. Müller (Garching, Woosley, S. E., & Weaver, T. A. 1995, ApJS, 101, 181 Germany: MPI für Astrophysik), 100 Zwerger, T., & Müller, E. 1997, A&A, 320, 209 A.Mareketal.:Animprovedeﬀective potential for supernova simulations, Online Material p 1

Online Material A.Mareketal.:Animprovedeﬀective potential for supernova simulations, Online Material p 2

2 Appendix A: Construction of an “effective” while the line element dsiso in radial isotropic coordinates is one-parameter equation of state given by The relativistic TOV solution for a spherical, self-gravitating 2 = −α2 2 + φ4 2 + 2 Ω2 . dsiso dtiso driso risod iso (B.2) matter distribution in equilibrium can be found by solving the following two coupled ordinary differential equations for the In vacuum outside the matter distribution, the metric compo- pressure P and the mass m: nents become −1 −1 2M 2M ∂P ρ(1 + ) + P 2m = − g , = − g = −1 = − m + 4πr3P 1 − , (A.1) A 1 B 1 A (B.3) ∂r r2 r rst rst ∂m and = 4πρ(1 + )r2. (A.2) ∂r M 1 − g M α = 2riso ,φ= + g , with appropriate boundary conditions at the inner and outer M 1 (B.4) = = 1 + g 2riso boundary at r 0andr R, respectively. Note that for a TOV 2riso solution the mass m(R) is equivalent to the total gravitational respectively. mass M of the matter configuration. g As with both metrics one can describe a spherical matter The above system is closed by choosing a two-parameter distribution in equilibrium, equating the line elements, ds2 = EoS P(ρ, ) and assuming a distribution profile for the specific st ds2 , yields conditions for each of the coordinates and metric internal energy ,whicheffectively transforms the EoS into a iso components in such a case. one-parameter relation, P = P(ρ) [or alternatively ρ = ρ(P)], Setting the two coordinate times equal, we obtain a condi- from which the specific internal energy can be determined as α = ,ρ tion for A and : (P ). A typical example for this is to demand a polytropic ⎫ 2 = α2 2 , ⎪ √ relation for P and to specify the internal energy by the ideal A dtst dt ⎬ iso ⎭⎪ −→ A = α. (B.5) gas EoS: dtst = dtiso, P 1/γ P = Kργ,ρ= , A similar condition can be obtained for the angle element: −→ K ⎫ (A.3) r2 dΩ2 = φ4r2 dΩ2 , ⎬⎪ P st st iso iso ⎪ −→ r = φ2r . (B.6) P = ρ(γ − 1),= · Ω = Ω , ⎭ st iso ρ(γ − 1) d st d iso For a two-parameter EoS of the form of Eqs. (12), (13), where From the radial elements, we get a relation for the radial differ- a priori no profile for the specific internal energy is known, entials: the following procedure can be applied to construct an “effec- √ = φ2 . tive” one-parameter EoS. For matter in equilibrium, e.g., at late B drst driso (B.7) times long after the core bounce phase, both the radial profiles Knowing the solution in the isotropic radial coordinate, it is ρ of the pressure P(r) and the density (r), as well as the spe- straightforward to derive a simple expression for r (r ) from st iso cific internal energy (r) can be assumed to be stationary. If the Eq. (B.6). pressure profile is monotonic, then for each value of P there On the other hand, obtaining the inverse relation riso(rst)is exists a unique location r. By inverting the pressure profile, more complicated. For this, we insert Eq. (B.6) into Eq. (B.7) −→ ρ P(r) r(P), we can thus relate a specific value (r)and (r) and arrive at to each value of P. This way a one-parameter EoS ρ = ρ(r(P)) √ = dr B r √ [and (r(P))] can be constructed. iso = = iso B. 2 (B.8) The excellent matching of the density profiles in the central drst φ rst part of a collapsed core in equilibrium obtained from a dynamic This differential equation can be solved by numerical integra- evolution with profiles from a TOV solution using equal cen- tion using, e.g., a Runge-Kutta integration scheme with the fol- tral density and the EoS transformed as described above (see lowing boundary conditions: Figs. 3 to 5) demonstrates the applicability of this method. r = r = 0, (B.9) 0iso 0st⎛ ⎞ ⎜ ⎟ Appendix B: Radial coordinate transformation R ⎜ M 2M ⎟ = st ⎜ − g + − g ⎟ · Riso ⎝⎜1 1 ⎠⎟ (B.10) To compare a solution of the TOV Eqs. (A.1), (A.2), which are 2 Rst Rst formulated in standard Schwarzschild-like coordinates, to results from our evolution code using relativistic gravity, which An alternative method is to rewrite Eq. (B.8) in terms of ln riso, √ is based on radial isotropic coordinates, we must perform a co- dlnr B ordinate transformation of the radial coordinate. iso = , (B.11) dr r In standard Schwarzschild-like coordinates, the line ele- st st 2 ment dsst reads which can be directly integrated numerically. 2 = − 2 + 2 + 2 Ω2 , dsst A dtst B drst rst d st (B.1)